Navier-Stokes — The Entropy-Bounded Blow-Up Criterion¶

April 2026 · Addendum to NS Global Regularity and NS v3 Paper

The Σ-reparameterization transforms the NS regularity question into a cleaner form: can the shellwise defect concentrate at a single entropy level?

The v3 Criterion (unchanged)¶

Shellwise High-Frequency Defect¶

\[\mathcal{D}_K(t) = \sum_{j \geq K}\left(\Pi_j(t) - \nu 2^{2j}|u_j(t)|_2^2\right)\]

Regularity Criterion¶

\[\sup_K \int_0^T \mathcal{D}_K^+(t)\,dt < \infty \implies \text{regularity}\]

Blow-up = flux outrunning dissipation at arbitrarily fine scales.

The Entropy-Time Criterion¶

Reparameterized Defect¶

\[\mathcal{D}_K(\Sigma) = \sum_{j \geq K}\left(\Pi_j(\Sigma) - \nu 2^{2j}|u_j(\Sigma)|_2^2\right)\]

where all time-dependent quantities are now functions of \(\Sigma\) via \(t(\Sigma)\).

Entropy-Time Regularity Criterion¶

\[\boxed{\sup_K \int_0^{\Sigma_T} \mathcal{D}_K^+(\Sigma)\,d\Sigma < \infty \implies \text{regularity}}\]

where \(\Sigma_T = \Sigma(T)\) is the entropy at time \(T\).

Why This Is Cleaner¶

1. Σ Is Bounded¶

The von Neumann entropy \(\Sigma = -\mathrm{Tr}(\rho_{PS}\ln\rho_{PS})\) is bounded above by \(\ln(\dim PS)\). If PS has finite effective dimension (which it does for any physical system), then:

\[0 \leq \Sigma \leq \Sigma_{\max} = \ln(\dim PS)\]

The entropy integral is over a bounded interval \([0, \Sigma_{\max}]\), not \([0, T]\) which can be \([0, \infty)\).

2. Blow-Up = Entropy Concentration¶

In clock-time, blow-up means \(\mathcal{D}_K^+\) grows without bound as \(t \to T^*\) (the blow-up time). In entropy-time, the question becomes: can \(\mathcal{D}_K^+(\Sigma)\) have a non-integrable singularity on the bounded interval \([0, \Sigma_{\max}]\)?

For the integral \(\int_0^{\Sigma_{\max}} \mathcal{D}_K^+(\Sigma)\,d\Sigma\) to diverge on a bounded interval, \(\mathcal{D}_K^+\) must blow up at least as \(\sim 1/(\Sigma^* - \Sigma)\) near some \(\Sigma^*\).

3. The Action-Entropy Connection¶

By the Action-Entropy Identity, the NS energy \(E = \frac{1}{2}\|u\|_2^2\) is related to \(\Sigma\) via the GL action. The standard energy estimate:

\[\frac{dE}{dt} = -\nu\|\nabla u\|_2^2\]

becomes:

\[\frac{dE}{d\Sigma} = -\frac{\nu}{\dot\Sigma}\|\nabla u\|_2^2\]

Energy dissipation per entropy unit depends on \(\dot\Sigma\). If \(\dot\Sigma\) is bounded below (which the monotonicity conjecture \(\dot\Sigma > 0\) implies), then energy dissipation per entropy unit is bounded above, constraining the blow-up rate.

The Entropy Blow-Up Question¶

The NS regularity problem in entropy-time reduces to:

Question: For smooth initial data \(u_0\) with finite energy, can the map \(\Sigma \mapsto \mathcal{D}_K(\Sigma)\) develop a non-integrable singularity on \([0, \Sigma_{\max}]\)?

Argument against blow-up (heuristic):

\(\Sigma\) is bounded above (\(\Sigma \leq \Sigma_{\max}\))
The GL action \(S_E = -\Sigma\) is bounded below (standard for \(\phi^4\) GL)
Therefore the energy density \(\rho_W\) is bounded above on the entropy interval
The shellwise defect \(\mathcal{D}_K\) is controlled by \(\rho_W\)
A bounded function on a bounded interval is integrable

The gap: Step 4 is the hard one. The defect \(\mathcal{D}_K\) involves the difference between nonlinear flux and linear dissipation. Bounding \(\rho_W\) bounds the total energy but doesn't directly bound the flux-dissipation imbalance at high frequencies.

The Dimensional Reduction Conjecture in Entropy-Time¶

The v3 conjecture states that NS regularity reduces to bounding the high-frequency defect. In entropy-time:

Entropy-time dimensional reduction: If \(\mathcal{D}_K(\Sigma)\) is uniformly bounded on \([0, \Sigma_{\max}]\) for all \(K\), then the solution is regular.

This is stronger than the clock-time version because the domain is bounded.

Confidence Update¶

v3: 54% confidence.

v4: 58% confidence. The entropy frame adds:

Bounded integration domain (\(\Sigma \leq \Sigma_{\max}\))
Energy-entropy coupling via \(S_E = -\Sigma\)
The GL boundedness-below gives \(\rho_W\) bounds for free
Cleaner formulation of the blow-up question

Remaining gap: Prove that \(\rho_W\) bounds control the shellwise defect at each shell \(K\). This is the nonlinear estimation problem.